A rhombus has one vertex at point P(−1, 2) and its diagonals intersect at the origin. If the length of one diagonal is 6 units, what are the possible coordinates for the other three vertices?

To find the possible coordinates for the other three vertices of the rhombus, we start with the known vertex P(-1, 2) and the given information about the diagonals intersecting at the origin (0, 0). Since the length of one diagonal is 6 units, this diagonal extends from a point (-3, 0) to point (3, 0) on the x-axis, making the midpoint the origin. Now let's find the coordinates of the other three vertices. The two remaining vertices must also lie on the other diagonal that runs perpendicular to the first diagonal and passes through the origin. The length of the other diagonal is unknown, but since it's a rhombus, the vertices will form right angles at the origin. The coordinates of the rhombus opposite vertex can be found by utilizing the midpoint formula and the relations of similar triangles or geometric properties of the rhombus. 1. One of the remaining vertices is at an equal distance from the origin but will be in the opposite direction (straight through the origin from P). Thus, we can create a second vertex at V1(1, -2). 2. The third vertex, V2, is found by using a rotation of the known vertex P around the origin (0, 0). Utilizing the 90-degree rotation transformation, we can find that the third vertex could be V2(2, 1). So, possible coordinates for the other three vertices of this rhombus are: 1. V1(1, -2) 2. V2(2, 1) 3. V3(-2, -1). Feel free to switch around the third vertex using different forms of transformation (like 180-degree rotation, etc.), providing you with even more variations!